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Equation 12 · Part 14 · Structure from Sequence: What Protein Folding Prediction Did and Did Not Settle

Denominator: displaystyleint e^-β F(x')dx'

p(x)=e−βF(x)∫e−βF(x′) dx′,β=1kBT,p(x) = \frac{e^{-\beta F(x)}}{\displaystyle\int e^{-\beta F(x')}\,dx'}, \qquad \beta = \frac{1}{k_{\mathrm B}T},
∫e−βF(x′) dx′\displaystyle\int e^{-\beta F(x')}\,dx'

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

displaystyleint e−e^-β F(x')dx' occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

A protein in solution is not a coordinate file. It is a distribution. Under equilibrium conditions the physically meaningful object is a Boltzmann-weighted ensemble over conformations, p(x)=e−βF(x)∫e−βF(x′) dx′,β=1kBTp(x) = \frac{e^{-\beta F(x)}}{\displaystyle\int e^{-\beta F(x')}\,dx'}, \qquad \beta = \frac{1}{k_{\mathrm B}T}. and function frequently depends on the populations and interconversion rates of its modes rather than on the single most populated one. A predicted structure is a point estimate. It carries no populations, no relative free energies, and no exchange kinetics — none of which can be recovered from the file by inspection.

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the article section

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